Any quasi-arithmetic mean of two variables has the balancing property .

The quasi-arithmetic mean is invariant with respect to offsets and scaling of :

Moreover, any of those properties is essentially sufficient to characterize quasi-arithmetic means; see Aczél-Dhombres, Chapter 17.

Indeed, the only homogeneous quasi-arithmetic means are the power means and the geometric mean; see Hardy-Littlewood-Pólya, page 68.